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PROFESSOR: OK, today's about
inverse functions, which is a

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new way to create one function
from another one.

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And the reason it's so important
is that we want to--

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this, the logarithm, is going to
be the inverse function for

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e to the x.

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We can't live without e to
the x on one side and the

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logarithm on the other side.

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So here's the idea of
inverse functions.

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Well, here are the
letters we use.

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Usually y is f of x.

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That's the standard letters.

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Then, for the inverse, I'm going
to use f with a minus

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one above the line.

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And notice though, it will
be x is f inverse of y.

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And let me show you
why that is.

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Let's just remember what
a function is.

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So function like f
is, I take an x--

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that's my input--

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then the function acts on that
input and produces an output.

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At some level, this is what a
function is, a bunch of inputs

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and the corresponding outputs.

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OK, what's the inverse
function?

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You can guess what's coming.

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I'll reverse those.

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For the inverse function,
y will be the input.

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So y is now the input.

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What used to be the output
is now the input--

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just turning them around.

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Then the question is, what
x did it come from?

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That x that y came from up here
is the x that it goes to

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with the inverse function.

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So you see the point?

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X and y are just getting
reversed.

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Let me do an example, because
that's letters, and we need a

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first example.

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y is x squared.

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So that's my function f of x.

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f is this squaring function.

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If you give me x equals three,
the output is y equals nine.

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Now, what's the inverse
function?

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The inverse function, I
want to find x from y.

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How do I do that?

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I take the square root.

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So the inverse function
will be x is--

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do you like to write square root
of y or do you like to

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write y to the one half power?

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Both good--

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That's the inverse function of
the other, and, of course,

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before we said, if x was
three y was nine.

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And, now, if y is nine, then x
will come out to be the square

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root of nine, three.

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Oh, one small point--
well, not so small.

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I was really staying there in
this example with x greater or

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equal zero.

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I don't know want to allow
x equal minus three.

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Well, why not?

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Because if I allowed x equal
minus three as one of the

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inputs, if I extended the
function x squared

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to go for all x's.

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So if x equal minus three was
allowed input, then y would be

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the same answer, nine.

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So I would be getting
nine from both

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three and minus three.

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And then, in the inverse,
I wouldn't know which

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one to go back to.

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In the inverse, the input would
be nine, but should the

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output be three or
minus three.

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So the point is, our functions
have to be--

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one-to-one is a kind of nice
expression that gives you the

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idea, one x for one y,
one y for one x.

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And that means that they're
graphs have to go steadily

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upwards or steadily downwards,
but not down and up the way y

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equal x squared would if
I went over all x's.

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Let me do some more examples
before I come to the reason

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for this lecture, which is the
exponential and the logarithm.

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And let's just look ahead to
what will come near the end of

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the lecture.

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We know facts about the
exponential, e to the x, and

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those facts, when I look at the
inverse function, give me

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some different facts, important
facts still, about

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the logarithm.

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And here is the most
important one.

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That the log of a product of two
numbers is the sum of the

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two logarithms, very
important fact.

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That simple but important
fact is what

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made logarithms famous.

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And it was the whole basis
for the slide rule.

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Well, do you know what
a slide rule is?

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Maybe you haven't
ever seen one?

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Probably not.

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Everybody had them, and then
suddenly nobody has any.

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But the point was, on a slide
rule you had a little stick,

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another little stick--

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I used to drop the thing--

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And you push out log y, and then
the second stick measures

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out log capital Y, and then you
read off the answer of a

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multiplication.

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So you were able to multiply,
but kind of inaccurately.

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So--

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But that doesn't mean that this

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logarithm isn't still important.

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It is, just not for
slide rules.

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I promised two more examples.

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Also this is a chance to
think about functions.

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So what about the radius of
a circle r and the area a?

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There is a function there.

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The input is r, and the
area is pi r squared.

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So that's some function of
r, input r, output a.

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Now, tell me the inverse
function.

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The inverse function, I'm
going to input a, and

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I want to get r.

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So the input for the inverse
function, it's

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going to be like this.

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I have to solve that
equation for r.

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How do I do that?

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I divide by pi.

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That gives me r squared.

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And then, just as there,
I take the square

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root, and I have r.

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So that's the inverse
function of--

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is that a function of r?

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No way.

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The input is now a.

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This is a function of a.

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Divide by pi, take the square
root, and you're back to r.

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Let me draw the pictures
that go with that.

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Because the graph of a function
and its inverse

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function are really
quite neat.

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So do you know what the
graph of a equals pi r

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squared would look like?

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Again, r is only going to be
positive, and, now, area's

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only going to be positive.

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And the graph of pi r squared
is a parabola.

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Say at r equal one, I
reach area equal--

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What would be the area
if the radius is one?

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Plug it in the formula,
the area is pi.

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That's that point.

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And those are all the
other points.

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OK.

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So that's the graph, which
was nothing new.

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The new graph is the graph
of the inverse function.

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OK.

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What's up?

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This time the input is now
a, and the output is r.

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If the area is 0,
the radius is 0.

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If the area is pi--

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Oh, look, I'm just going
to take this, and

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it's going to go here.

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If the area is pi, what's
the radius?

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Well, put it in the formula.

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If the area is pi, I have
pi over pi, one,

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square root's one.

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The radius is one, of
course it's one.

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One there, so that's a point
on the graph of the inverse

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function, of this square
root of a over pi.

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And what's the rest
of the graph.

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Does it look like that?

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No way.

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Everything is being flipped,
you could say, Or you could

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say, a mirror image just
turned over here.

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This thing, which started out
like this, is now a square

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root function.

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The square root function climbs
and comes around like

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that like it's a parabola this
way, because that one was a

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parabola that way.

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All right, let me go on
to a second example.

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But that point that these graphs
just flip over the 45

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degree line, it's because y and
x are getting switched.

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Let me do the second example.

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What about temperature?

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We could measure temperature
in Fahrenheit, say f.

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Or we can measure it in
centigrade or Celsius, say c.

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And what's the function?

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If I take f, I want to know c.

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The centigrade temperature
is some function of f.

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Let me at the same
time draw the

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picture so we can remember.

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So this is now the
forward function.

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I'm creating f first, and then
I'm going to create f inverse.

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OK.

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Do you remember the point
how they're connected?

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Here is f.

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And we'll start with the
freezing point of water.

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The freezing point of water is
32 Fahrenheit but is zero

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centigrade.

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That's why that system
got created.

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So f equals 32, c equals zero.

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That's on the graph.

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And then what's the
other key point?

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The boiling point of water, so
say, that's one f is 212.

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212 is the boiling point
of water in Fahrenheit.

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And what's the boiling point
in Celsius, centigrade?

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100--

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I mean that system was--

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I don't know where 32 and 212
came from, but 0 to 100 is

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pretty sensible,
0 and then 100.

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So that's the other point and
then, actually, the graph is

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just straight line.

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In fact, let's find
the formula.

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What's the equation
for that line?

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So I take f and I subtract 32,
so that gets me at the right

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start, the right
freezing point.

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And now I want to multiply by
the right slope to get up to

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the right boiling point.

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So when I go over 180,
I want to go up 100.

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So it's 100 over 180.

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That 180 was the 32 to 212.

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So the ratio of 100 to 180
that's, well, five to nine

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would be easier to write, so
let me write five to nine.

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Is that OK for the graph of
the original function?

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This is my function
of f giving me c.

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Ready for the inverse
function?

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Can you do this with me?

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c is now the input.

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f is now the output.

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What was c equals 0 and
a 100, those were the

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key points for c.

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f equals 32 and 212 were
they key points for f.

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This was on the graph, right?

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0 centigrade gives
32 Fahrenheit.

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100 centigrade matches 212.

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That's on the graph.

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And again, it's a
line in between.

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Hoo!

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My picture isn't so fantastic.

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That 212 really should be
higher, and that line should

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be steeper.

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Let's see that from the
formula for f inverse.

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What is going to be the
steepness of the second line?

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OK.

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00:14:24,575 --> 00:14:29,300

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Here I've done graphs
with some numbers.

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Here I'm going to do algebra.

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I mean, the point
of algebra is--

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you may have wondered, what
was the point of algebra--

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the point is to deal with
all numbers at once.

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00:14:48,285 --> 00:14:50,860

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00:14:50,860 --> 00:14:55,020
I could write down some other
numbers like, some in between

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number like 122 or something,
probably corresponds to a

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centigrade of 50.

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But I can't live forever
with numbers.

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I need symbols.

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That's where letters,
algebra, comes in.

259
00:15:13,560 --> 00:15:15,770
So now, I'm going
to do algebra.

260
00:15:15,770 --> 00:15:20,960
I want to get Fahrenheit
out of centigrade.

261
00:15:20,960 --> 00:15:23,940
I want to solve this
equation for f.

262
00:15:23,940 --> 00:15:25,160
How do you solve for f?

263
00:15:25,160 --> 00:15:30,920
Well, first thing is, get rid of
that 5/9, multiply by 9/5.

264
00:15:30,920 --> 00:15:35,280
So now I have 9/5 of c.

265
00:15:35,280 --> 00:15:38,480
So that 5/9 is now over here.

266
00:15:38,480 --> 00:15:40,330
Now, I have an f minus 32.

267
00:15:40,330 --> 00:15:43,500
I want to bring the 32 over
on to the c side.

268
00:15:43,500 --> 00:15:44,885
It'll come over as a plus.

269
00:15:44,885 --> 00:15:48,630

270
00:15:48,630 --> 00:15:52,980
So I've solved this equation
for f, and that's told me,

271
00:15:52,980 --> 00:15:54,340
what's the inverse function.

272
00:15:54,340 --> 00:15:56,620
And you notice, it is
a straight line.

273
00:15:56,620 --> 00:16:00,010
And what's it's slope
by the way?

274
00:16:00,010 --> 00:16:05,630
Its slope is 9/5, where this
had a slope of 5/9.

275
00:16:05,630 --> 00:16:10,120
That's going to happen,
if you multiply.

276
00:16:10,120 --> 00:16:14,570
And sooner or later, in the
inverse, you have to divide.

277
00:16:14,570 --> 00:16:18,930
So one slope is the reciprocal
of the other slope.

278
00:16:18,930 --> 00:16:21,670
Well, it's especially easy when
we see straight lines.

279
00:16:21,670 --> 00:16:23,860
OK.

280
00:16:23,860 --> 00:16:29,670
Now, are we ready for the real
thing, meaning exponentials?

281
00:16:29,670 --> 00:16:31,170
OK.

282
00:16:31,170 --> 00:16:35,830
So I come back to this board,
which tells me what I'm after.

283
00:16:35,830 --> 00:16:40,280
And raise that a little
and go for it.

284
00:16:40,280 --> 00:16:43,880

285
00:16:43,880 --> 00:16:46,360
So, what am I saying here?

286
00:16:46,360 --> 00:16:51,950
I'm saying that the logarithm
is going to be the inverse

287
00:16:51,950 --> 00:16:56,310
function of e to the x.

288
00:16:56,310 --> 00:16:59,510
And it's called the natural
logarithm, and we use this

289
00:16:59,510 --> 00:17:03,055
letter n for natural.

290
00:17:03,055 --> 00:17:06,170

291
00:17:06,170 --> 00:17:10,170
Although, the truth is, that
it's the only logarithm

292
00:17:10,170 --> 00:17:11,900
I ever think of.

293
00:17:11,900 --> 00:17:18,349
I would freely write L-O-G,
because I would always mean

294
00:17:18,349 --> 00:17:20,640
this natural logarithm.

295
00:17:20,640 --> 00:17:25,450

296
00:17:25,450 --> 00:17:29,600
So I'm defining it as the
inverse and probably a graph

297
00:17:29,600 --> 00:17:35,560
is the way to see what
it looks like.

298
00:17:35,560 --> 00:17:39,570
So I need to graph of e
to the x, and then, a

299
00:17:39,570 --> 00:17:40,660
graph of its inverse.

300
00:17:40,660 --> 00:17:45,870
And then, by the way, since
we're doing calculus, our next

301
00:17:45,870 --> 00:17:50,340
lecture is going to
find derivatives.

302
00:17:50,340 --> 00:17:52,920
We know the derivative
of e to the x.

303
00:17:52,920 --> 00:17:54,340
It's e to the x.

304
00:17:54,340 --> 00:18:03,440
That's the remarkable property
that we started with.

305
00:18:03,440 --> 00:18:07,280
Then we'll find the
derivative of the

306
00:18:07,280 --> 00:18:10,010
log, the inverse function.

307
00:18:10,010 --> 00:18:14,060
And it will come out to be
remarkable too, amazing,

308
00:18:14,060 --> 00:18:17,680
amazing, just what we
needed, in fact.

309
00:18:17,680 --> 00:18:20,640
All right, but let's get an idea
what that log looks like.

310
00:18:20,640 --> 00:18:25,070
I know you've seen logs before,
but now we have this

311
00:18:25,070 --> 00:18:30,890
base e, e to the x that only
comes in calculus.

312
00:18:30,890 --> 00:18:35,980
And let's graph it.

313
00:18:35,980 --> 00:18:41,810
So now my function of x
is e to the x, and I

314
00:18:41,810 --> 00:18:43,060
want to graph it.

315
00:18:43,060 --> 00:18:46,070

316
00:18:46,070 --> 00:18:49,380
This is, of course, y.

317
00:18:49,380 --> 00:18:57,220
Actually, I realize, x
can be negative or

318
00:18:57,220 --> 00:19:00,760
positive, no problem.

319
00:19:00,760 --> 00:19:02,530
But y--

320
00:19:02,530 --> 00:19:05,810
e to the x, always comes
out positive.

321
00:19:05,810 --> 00:19:09,090
The graph is going
to be above--

322
00:19:09,090 --> 00:19:11,120
Here's x.

323
00:19:11,120 --> 00:19:14,560
Let me draw the graph from
0 to one and say,

324
00:19:14,560 --> 00:19:15,810
back to minus one.

325
00:19:15,810 --> 00:19:18,980

326
00:19:18,980 --> 00:19:25,040
Then the graph is going to
be above the axis here.

327
00:19:25,040 --> 00:19:26,760
Let's see, where is it?

328
00:19:26,760 --> 00:19:30,140

329
00:19:30,140 --> 00:19:33,240
When x is 0, what's y?

330
00:19:33,240 --> 00:19:39,970
y is e to the 0th power,
which is one.

331
00:19:39,970 --> 00:19:41,890
e to the 0 is one.

332
00:19:41,890 --> 00:19:45,880
The exponential function starts
at one, right there.

333
00:19:45,880 --> 00:19:48,960
That height is one.

334
00:19:48,960 --> 00:19:57,210
Now when x is one, y is e to the
first power, which is e,

335
00:19:57,210 --> 00:19:59,880
about 2.78.

336
00:19:59,880 --> 00:20:05,000
So maybe up there, somewhere
about here.

337
00:20:05,000 --> 00:20:10,600
So that height is e,
corresponding to one.

338
00:20:10,600 --> 00:20:14,520
And what about when
x is minus one?

339
00:20:14,520 --> 00:20:19,160
Then y is e to the minus one.

340
00:20:19,160 --> 00:20:21,650
e to the minus one is--

341
00:20:21,650 --> 00:20:24,340
that minus says, divide.

342
00:20:24,340 --> 00:20:29,370
It's one over e to the first
power, one over 2.78,

343
00:20:29,370 --> 00:20:33,330
something like 1/3 or so,
something about there.

344
00:20:33,330 --> 00:20:41,290
And now, if I put in the
other points here, the

345
00:20:41,290 --> 00:20:44,225
graph looks like that.

346
00:20:44,225 --> 00:20:48,220
And, actually, the reason I
didn't go beyond x equal one

347
00:20:48,220 --> 00:20:58,450
is that it climbs so fast.
e to the x takes off.

348
00:20:58,450 --> 00:21:04,990
It grows exponentially, if you
can allow me to say that.

349
00:21:04,990 --> 00:21:08,710
Which reminds me, we don't
often say, grows

350
00:21:08,710 --> 00:21:11,340
logarithmically.

351
00:21:11,340 --> 00:21:14,320
Well, let's see what grows
logarithmically means.

352
00:21:14,320 --> 00:21:16,770
It means creeping along.

353
00:21:16,770 --> 00:21:22,220
If e to the x is zipping up
real fast, then the log is

354
00:21:22,220 --> 00:21:24,550
going to go up only slowly.

355
00:21:24,550 --> 00:21:28,230

356
00:21:28,230 --> 00:21:29,720
So I want the inverse
function.

357
00:21:29,720 --> 00:21:33,060
Of course, this graph
continues,

358
00:21:33,060 --> 00:21:35,250
gets very, very small.

359
00:21:35,250 --> 00:21:36,830
It continues up here.

360
00:21:36,830 --> 00:21:42,380
It gets very, very big
but keeps going.

361
00:21:42,380 --> 00:21:48,700
Now, ready for x equals log y.

362
00:21:48,700 --> 00:21:55,130
And remember, I'm going to
draw its picture, and I'm

363
00:21:55,130 --> 00:22:01,430
defining that function as the
inverse function of the one we

364
00:22:01,430 --> 00:22:04,940
have. So I'm not going
to give a new

365
00:22:04,940 --> 00:22:07,750
definition, a new function.

366
00:22:07,750 --> 00:22:17,470
It's defined by being the
inverse function.

367
00:22:17,470 --> 00:22:19,460
That's what it is.

368
00:22:19,460 --> 00:22:23,370
But now we know, from experience
with two graphs, we

369
00:22:23,370 --> 00:22:26,170
know what its graph is
going to look like.

370
00:22:26,170 --> 00:22:29,130
So x is now going to be
this graph, and y is

371
00:22:29,130 --> 00:22:29,730
going to be that one.

372
00:22:29,730 --> 00:22:32,150
So y only is positive.

373
00:22:32,150 --> 00:22:35,910
We can only take the log
of positive numbers.

374
00:22:35,910 --> 00:22:38,680

375
00:22:38,680 --> 00:22:41,270
The log of a negative number,
that's something imaginary,

376
00:22:41,270 --> 00:22:42,520
we're not touching that.

377
00:22:42,520 --> 00:22:45,400

378
00:22:45,400 --> 00:22:55,990
The log can come out positive,
zero, negative.

379
00:22:55,990 --> 00:22:57,150
x could be anything.

380
00:22:57,150 --> 00:23:00,290
Here is x.

381
00:23:00,290 --> 00:23:02,630
Let's put in the
points we know.

382
00:23:02,630 --> 00:23:08,020
They'll be the same three points
as here, but you see

383
00:23:08,020 --> 00:23:15,210
that x axis is now vertical, the
y axis is now horizontal.

384
00:23:15,210 --> 00:23:18,330
I put in these points,
now let me put in--

385
00:23:18,330 --> 00:23:21,050
So what's the thing?

386
00:23:21,050 --> 00:23:26,370
When y is 0, what's x?

387
00:23:26,370 --> 00:23:28,490
Yeah, can you get that one?

388
00:23:28,490 --> 00:23:29,740
What's the log--oh, no.

389
00:23:29,740 --> 00:23:32,620

390
00:23:32,620 --> 00:23:34,690
y doesn't make it to 0.

391
00:23:34,690 --> 00:23:38,020
When y is one, that's
what I meant to say.

392
00:23:38,020 --> 00:23:43,250
When y is one, what's
the log of one?

393
00:23:43,250 --> 00:23:44,810
What is the log of one?

394
00:23:44,810 --> 00:23:47,730
That's a key point here,
and we see it.

395
00:23:47,730 --> 00:23:50,580
We got y equal one
when x is 0.

396
00:23:50,580 --> 00:23:56,930
The logarithm of one is--

397
00:23:56,930 --> 00:23:59,980
so when y is one--

398
00:23:59,980 --> 00:24:01,540
is that right?

399
00:24:01,540 --> 00:24:05,170
Logarithm of one, let's
put in that one.

400
00:24:05,170 --> 00:24:07,410
The logarithm of one is 0.

401
00:24:07,410 --> 00:24:10,150
That's a point on our curve.

402
00:24:10,150 --> 00:24:11,770
That's a point on our curve.

403
00:24:11,770 --> 00:24:16,690
This point flips down
to this point.

404
00:24:16,690 --> 00:24:20,620
Can I just remind myself,
because you saw me hesitating,

405
00:24:20,620 --> 00:24:23,590
that the log of one is 0.

406
00:24:23,590 --> 00:24:26,910
It's nice to have a
couple of numbers.

407
00:24:26,910 --> 00:24:28,460
Then what are the other
ones I want?

408
00:24:28,460 --> 00:24:31,260
I want to know the log
of e, and I want to

409
00:24:31,260 --> 00:24:34,980
know the log of 1/e.

410
00:24:34,980 --> 00:24:38,570
And what are those logarithms?

411
00:24:38,570 --> 00:24:41,160
I could look over here.

412
00:24:41,160 --> 00:24:44,860
They're going to be
one and minus one.

413
00:24:44,860 --> 00:24:49,000
But let's just begin to get
the idea of the log.

414
00:24:49,000 --> 00:24:54,390
The log is the exponent.

415
00:24:54,390 --> 00:24:56,860
That's what you should say
to yourself all the time.

416
00:24:56,860 --> 00:24:58,290
What is the logarithm?

417
00:24:58,290 --> 00:25:03,360
The logarithm is the exponent
in the original.

418
00:25:03,360 --> 00:25:08,510
So here the exponent is one,
so the log is one.

419
00:25:08,510 --> 00:25:10,930
What's the exponent there?

420
00:25:10,930 --> 00:25:14,480
One over e, that's e to
the minus one power.

421
00:25:14,480 --> 00:25:18,260
That's log of e to the
minus one power.

422
00:25:18,260 --> 00:25:19,890
And what is that logarithm?

423
00:25:19,890 --> 00:25:23,260
It's the exponent minus one.

424
00:25:23,260 --> 00:25:25,030
Let me plot those points.

425
00:25:25,030 --> 00:25:27,030
Here is e.

426
00:25:27,030 --> 00:25:30,350
So there is one, here
is e, here is 1/e.

427
00:25:30,350 --> 00:25:34,760

428
00:25:34,760 --> 00:25:36,750
The logarithm of one was 0.

429
00:25:36,750 --> 00:25:38,430
That point's on my graph.

430
00:25:38,430 --> 00:25:40,800
The logarithm of e is one.

431
00:25:40,800 --> 00:25:42,750
This point's on my graph.

432
00:25:42,750 --> 00:25:45,760
The logarithm of 1/e
is minus one.

433
00:25:45,760 --> 00:25:55,040
The curve is coming up like that
but bending down just the

434
00:25:55,040 --> 00:25:59,420
way this curve was bending up.

435
00:25:59,420 --> 00:26:03,970
And if I continue the curve, the
logarithm would get more

436
00:26:03,970 --> 00:26:04,720
and more negative.

437
00:26:04,720 --> 00:26:11,540
It's headed down there, but y
is never allowed to be 0.

438
00:26:11,540 --> 00:26:14,320
Headed up here, what happens?

439
00:26:14,320 --> 00:26:18,450
The log of a million, the
log of a trillion--

440
00:26:18,450 --> 00:26:22,960
I mean, we can deal with the
national debt, just take its

441
00:26:22,960 --> 00:26:31,260
log, because it climbs
so slowly.

442
00:26:31,260 --> 00:26:32,770
Notice it kept climbing.

443
00:26:32,770 --> 00:26:37,700
It doesn't peak off here.

444
00:26:37,700 --> 00:26:40,630
That's a little farther than
I intended to draw it.

445
00:26:40,630 --> 00:26:45,090
That's pretty far out on the
y axis, but not very

446
00:26:45,090 --> 00:26:46,420
high on the x axis.

447
00:26:46,420 --> 00:26:50,810
Logarithms of big numbers
are quite small numbers.

448
00:26:50,810 --> 00:26:56,680
And that's actually
why, as we'll see,

449
00:26:56,680 --> 00:26:58,490
people use log paper.

450
00:26:58,490 --> 00:27:00,640
They draw a log-log graphs.

451
00:27:00,640 --> 00:27:04,570
That's to get big numbers
on to the graph by

452
00:27:04,570 --> 00:27:06,480
dealing with logs.

453
00:27:06,480 --> 00:27:12,210
So that's what I want to say
about the logarithm.

454
00:27:12,210 --> 00:27:20,070
Except, to come back to
these two key facts,

455
00:27:20,070 --> 00:27:23,010
especially the first one.

456
00:27:23,010 --> 00:27:26,780
Can I find space for
that first one?

457
00:27:26,780 --> 00:27:32,990

458
00:27:32,990 --> 00:27:36,130
So y is e to the x, as always.

459
00:27:36,130 --> 00:27:46,770
Capital Y would be e to the
capital X. And now, the

460
00:27:46,770 --> 00:27:50,450
interesting property is what
happens if I multiply.

461
00:27:50,450 --> 00:27:53,010
What happens if I multiply
y times Y?

462
00:27:53,010 --> 00:27:55,870

463
00:27:55,870 --> 00:28:01,780
I have, that's e to the x times
e to the X. That's what

464
00:28:01,780 --> 00:28:04,210
the little y and big y were.

465
00:28:04,210 --> 00:28:09,930
Now we're ready to use the
crucial property of the

466
00:28:09,930 --> 00:28:11,180
exponential curve.

467
00:28:11,180 --> 00:28:13,660

468
00:28:13,660 --> 00:28:16,670
I'm asking you, because
you have to know this.

469
00:28:16,670 --> 00:28:19,905
What is e to the x times
e to the capital X?

470
00:28:19,905 --> 00:28:25,520

471
00:28:25,520 --> 00:28:29,440
Suppose x was two and
capital X was three?

472
00:28:29,440 --> 00:28:35,590
Then I have e times e, e
squared, multiplying e times e

473
00:28:35,590 --> 00:28:39,570
times e, three e's.

474
00:28:39,570 --> 00:28:40,370
What do I have?

475
00:28:40,370 --> 00:28:45,520
I've got e times e, time
e times e time e.

476
00:28:45,520 --> 00:28:47,715
All together five e's are
getting multiplied.

477
00:28:47,715 --> 00:28:50,420

478
00:28:50,420 --> 00:28:55,440
I just add the exponents.

479
00:28:55,440 --> 00:28:58,870
That's the big rule for
the exponential.

480
00:28:58,870 --> 00:29:04,360
If I multiply exponentials,
I add the exponents.

481
00:29:04,360 --> 00:29:09,040
Now, I just want to convert that
to a rule for logarithms.

482
00:29:09,040 --> 00:29:10,950
I'm going to do the
inverse function.

483
00:29:10,950 --> 00:29:14,990
I'm going to take the log of
both sides, and, I hope, we're

484
00:29:14,990 --> 00:29:16,620
going to get the right thing.

485
00:29:16,620 --> 00:29:22,790
The logarithm of this is--

486
00:29:22,790 --> 00:29:27,910
Well, what's the logarithm
of this result?

487
00:29:27,910 --> 00:29:30,610
It's the exponent.

488
00:29:30,610 --> 00:29:34,820
The logarithm of this
number is that.

489
00:29:34,820 --> 00:29:39,120
Just the way the logarithm of e
to that number was the one.

490
00:29:39,120 --> 00:29:42,260
The logarithm of e to that
number was the minus one.

491
00:29:42,260 --> 00:29:47,470
The logarithm of this number is
the exponent x plus capital

492
00:29:47,470 --> 00:29:53,380
X. And finally what
is little x?

493
00:29:53,380 --> 00:29:55,740
Well, don't forget where
it came from.

494
00:29:55,740 --> 00:30:02,480
Little x is the exponent for
y, so little x is log y.

495
00:30:02,480 --> 00:30:09,810
And capital X is
log capital Y.

496
00:30:09,810 --> 00:30:12,900
Bunch of symbols
on that board.

497
00:30:12,900 --> 00:30:16,670
And the last line is the one
that we were shooting for.

498
00:30:16,670 --> 00:30:21,730
The logarithm of y times Y
is the sum of the logs.

499
00:30:21,730 --> 00:30:25,360

500
00:30:25,360 --> 00:30:27,585
Because this guy is
also important--

501
00:30:27,585 --> 00:30:31,940

502
00:30:31,940 --> 00:30:35,570
Maybe I don't even
give a proof.

503
00:30:35,570 --> 00:30:39,150
Because it's intimately related
to this one, why don't

504
00:30:39,150 --> 00:30:40,720
I just see it.

505
00:30:40,720 --> 00:30:42,545
What would be the log
of y squared?

506
00:30:42,545 --> 00:30:51,370

507
00:30:51,370 --> 00:30:53,650
Actually, we already-- here.

508
00:30:53,650 --> 00:30:57,750
If I wanted little y squared,
what should I do?

509
00:30:57,750 --> 00:31:00,110
I can get that answer
from what I've done.

510
00:31:00,110 --> 00:31:04,660
The log of little y squared, I
just take big Y to be the same

511
00:31:04,660 --> 00:31:08,180
as little y, I take big X to be
the same as little x, and

512
00:31:08,180 --> 00:31:10,190
I've got the log of y squared.

513
00:31:10,190 --> 00:31:14,030
Then is x plus x, two x's--

514
00:31:14,030 --> 00:31:16,910

515
00:31:16,910 --> 00:31:18,810
but x is the log of y.

516
00:31:18,810 --> 00:31:23,660

517
00:31:23,660 --> 00:31:29,530
If you square a number, you
only double its log.

518
00:31:29,530 --> 00:31:37,860
You're again seeing why these
numbers can grow very quickly

519
00:31:37,860 --> 00:31:41,710
by squaring and squaring and
squaring, but the logarithms

520
00:31:41,710 --> 00:31:46,940
only grow by multiplying by
two, only going up slowly.

521
00:31:46,940 --> 00:31:54,150
And then the general result
would be for any power, not

522
00:31:54,150 --> 00:31:57,840
just n equals two, not just n
equals a whole number, not

523
00:31:57,840 --> 00:32:05,450
just n equals positive numbers,
but all n, will be--

524
00:32:05,450 --> 00:32:07,670
I'll have n of these--

525
00:32:07,670 --> 00:32:11,470
so I'll have n logarithm of y.

526
00:32:11,470 --> 00:32:19,060
So that's a closely related
property that takes the same y

527
00:32:19,060 --> 00:32:19,740
to different powers.

528
00:32:19,740 --> 00:32:20,990
OK.

529
00:32:20,990 --> 00:32:23,720

530
00:32:23,720 --> 00:32:29,170
Lots of symbols today, but you
had to get that logarithm

531
00:32:29,170 --> 00:32:32,420
function straight before we
can take its derivative.

532
00:32:32,420 --> 00:32:34,800
Can I tell you what
its derivative is?

533
00:32:34,800 --> 00:32:37,410
Would you like to
know in advance?

534
00:32:37,410 --> 00:32:38,660
The derivative--

535
00:32:38,660 --> 00:32:41,460

536
00:32:41,460 --> 00:32:43,080
I don't know if I
should tell you.

537
00:32:43,080 --> 00:32:49,160
The derivative of log y, the
derivative of this log

538
00:32:49,160 --> 00:32:52,600
function, turns out to be 1/y.

539
00:32:52,600 --> 00:32:56,730

540
00:32:56,730 --> 00:32:58,360
Isn't that nice.

541
00:32:58,360 --> 00:33:01,950
A really good answer coming
from this function that we

542
00:33:01,950 --> 00:33:04,040
created as an inverse
function.

543
00:33:04,040 --> 00:33:06,750
And I'll just say here that now

544
00:33:06,750 --> 00:33:09,140
we've created the function.

545
00:33:09,140 --> 00:33:11,220
We've got it.

546
00:33:11,220 --> 00:33:15,680
Then I don't mind if you give it
a different letter, give it

547
00:33:15,680 --> 00:33:16,850
another name.

548
00:33:16,850 --> 00:33:19,590
Well, I hope you keep
its name log.

549
00:33:19,590 --> 00:33:21,190
Most people use that name.

550
00:33:21,190 --> 00:33:22,810
But you could use a
different letter.

551
00:33:22,810 --> 00:33:26,910
I'm perfectly happy for you to
write this as the derivative

552
00:33:26,910 --> 00:33:31,510
of log x is 1/x.

553
00:33:31,510 --> 00:33:35,340
Between that and that, I've
just changed letters.

554
00:33:35,340 --> 00:33:40,820
That was like after the real
thinking of this lecture,

555
00:33:40,820 --> 00:33:45,530
which was the when x was an
input and y was an output, and

556
00:33:45,530 --> 00:33:48,610
I really needed two
different letters.

557
00:33:48,610 --> 00:33:50,120
OK, good, that's inverse
functions.

558
00:33:50,120 --> 00:33:52,226
Thank you.

559
00:33:52,226 --> 00:33:54,480
ANNOUNCER: This has been
a production of MIT

560
00:33:54,480 --> 00:33:56,870
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Gilbert Strang.

561
00:33:56,870 --> 00:33:59,150
Funding for this video was
provided by the Lord

562
00:33:59,150 --> 00:34:00,360
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563
00:34:00,360 --> 00:34:03,490
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564
00:34:03,490 --> 00:34:06,570
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565
00:34:06,570 --> 00:34:08,130
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566
00:34:08,130 --> 00:34:10,274